Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations

Robert J. Elliott; Tak Kuen Siu

doi:10.3934/dcdsb.2017003

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2017, Volume 22, Issue 1: 59-81. Doi: 10.3934/dcdsb.2017003

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Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations

Robert J. Elliott^1,2,3, , and
Tak Kuen Siu^4,

1.
School of Mathematical Sciences, University of Adelaide, Adelaide, South Australia, Australia
2.
Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada
3.
Centre for Applied Financial Studies, University of South Australia, Adelaide, South Australia, Australia
4.
Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

* Corresponding author: Robert J. Elliott
* Corresponding author: Robert J. Elliott

Received: December 31, 2015

Revised: May 31, 2016

Published: November 2017

This paper is dedicated to Professor K.L. Teo for his 70$^{th}$ birthday

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Abstract

This paper considers a new stochastic volatility model with regime switches and uncertain noise in discrete time and discusses its theoretical development for filtering and estimation. The model incorporates important features for asset price models, such as stochastic volatility, regime switches and parameter uncertainty in Gaussian noises for both the return and volatility processes. In particular, both drift and volatility uncertainties for the return and volatility processes are incorporated by introducing a family of real-world probability measures. Then, by modifying the reference probability approach to filtering, a sequence of conditional sub-linear expectations is used to provide a robust approach for describing the drift and volatility uncertainties in the Gaussian noises. Filtering theory, based on conditional sublinear expectations and the Viterbi algorithm are adopted to derive filters for the hidden Markov chain and filter-based estimates of the unknown parameters.

Keywords:

Stochastic volatility,
hidden Markov models,
conditional sub-linear expectations,
filtering,
modified reference probability approach,
drift and volatility uncertainties.

Mathematics Subject Classification: Primary:91G70, 93E11;Secondary:91G80.

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